
As we explained at the beginning of this section, we consider
subalgebras of direct products of finitely many different algebras
rather than just subalgebras of finite direct powers
of the same algebra.
In addition, we don't assume that the algebras are finite,
therefore we use a less restrictive notion of a fork than
the notion used in [1]}, cf. Remark .

Given \(e\) , we produce a \(P\) -term \(T_n\)  for every \(n\) ,
which describes a uniform way for obtaining
each element of a \(P\) -subalgebra \(\mathbf {B}\)  of
a product \(\mathbf {A}_1\times \dots \times \mathbf {A}_n\)  in one step
from members of an arbitrary \((\mathsf {d},e)\) -representation for \(\mathbf {B}\) .
Since we are interested in algorithmic applications,
the advantage of finding these terms \(T_n\)  explicitly is that —
via the minimal circuit representation of \(T_n\)  — we can
get a useful upper bound on the length of computations needed to generate
elements of \(\mathbf {B}\)  from
members of a \((\mathsf {d},e)\) -representation for \(\mathbf {B}\) .

